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线性数学 Linear Mathematics MATH1007

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这是一份nottingham诺丁汉大学MATH1007作业代写的成功案例

线性数学 Linear Mathematics MATH1007
问题 1.

$$
\hat{L} \hat{U}=\hat{P}(A+\delta A) \quad \text { with }|\delta A| \leq \frac{2 n \epsilon}{1-n \epsilon}|\hat{L}||\hat{U}|
$$
and for the particular case that $m=n$ and $A$ is nonsingular, if an approximate solution, $\hat{\mathbf{x}}$, to $A \mathbf{x}=\mathbf{b}$ is computed by solving the two triangular linear systems, $\hat{L} \mathbf{y}=\hat{P} \mathbf{b}$ and $\hat{U} \hat{\mathbf{x}}=\mathbf{y}$, then $\hat{\mathbf{x}}$ is the exact solution to a perturbed linear system:

证明 .

$$
(A+\delta A) \hat{\mathbf{x}}=\mathbf{b} \quad \text { with } \quad|\delta A| \leq \frac{2 n \epsilon}{1-n \epsilon} \hat{P}^{T}|\mathcal{L}||\hat{U}|
$$
Furthermore, $\left|L_{i, j}\right| \leq 1$ and $\left|U_{i, j}\right| \leq 2^{i-1} \max {k \leq i}\left|A{k, j}\right|$, so
$$
|\delta A|_{\infty} \leq \frac{2^{n} n^{2} \epsilon}{1-n \epsilon}|A|_{\infty}
$$

英国论文代写Viking Essay为您提供作业代写代考服务

MATH1007 COURSE NOTES ：

$$
\hat{G} \hat{G}^{T}=A+\delta A \quad \text { with } \quad|\delta A| \leq \frac{(n+1) \epsilon}{1-(n+1) \epsilon}|\hat{G}|\left|\hat{G}^{T}\right| .
$$
Furthermore, if an approximate solution, $\hat{\mathbf{x}}$, to $A \mathbf{x}=\mathbf{b}$ is computed by solving the two triangular linear systems $\hat{G} \mathbf{y}=\mathbf{b}$ and $\hat{G}^{T} \hat{\mathbf{x}}=\mathbf{y}$, and a scaling matrix is defined as $\Delta=\operatorname{diag}\left(\sqrt{a_{i i}}\right)$, then the scaled error $\Delta(\mathbf{x}-\hat{\mathbf{x}})$ satisfies
$$
\frac{|\Delta(\mathbf{x}-\hat{\mathbf{x}})|_{2}}{|\Delta \mathbf{x}|_{2}} \leq \frac{\kappa_{2}(H) \epsilon}{1-\kappa_{2}(H) \epsilon}
$$








线性数学| Linear Mathematics代写 MT2501

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这是一份andrews圣安德鲁斯大学 MT2501作业代写的成功案例

实践统计学| Statistics in Practice代写 MT1007
问题 1.

where $Z_{i}$ is the coordination number of site $i$. We now introduce a site-site Green function $G_{i j}$ defined by
$$
Z_{i} G_{i k}-\sum_{j} G_{j k}=-\delta_{i k},
$$
where, physically, $G_{i j}$ is the field at $j$ as a result of injecting a unit flux at $i$. With the aid of the Green function, Eq. (35) becomes
$$
P_{i}=P_{i}^{0}+\sum_{j} \sum_{k} G_{i j} \Delta_{j k}\left(P_{j}-P_{k}\right)
$$


证明 .

$$
Q_{i j}=Q_{i j}^{0}+\sum_{[l k]} Q_{l k}\left(G_{i l}+G_{j k}-G_{j l}-G_{i k}\right),
$$
where $[l k]$ indicates that the bond connecting nearest-neighbor sites $l$ and $k$ is counted only once in the sum. We denote bonds with Greek letters and assign direction to them and let
$$
\gamma_{\alpha \beta}=\left(G_{i l}+G_{j k}\right)-\left(G_{j l}+G_{i k}\right),
$$


英国论文代写Viking Essay为您提供作业代写代考服务

MT2501 COURSE NOTES ：

The MG approximation can be generalized to macroscopically-anisotropic materials that consist of $n-1$ different types of unidirectionally aligned isotropic inclusions of the same shape, and is given by
$$
\sum_{i=1}^{n} \phi_{i}\left(\mathbf{C}{e}-\mathbf{C}{1}\right) \cdot \mathbf{T}{i}^{(1)}=\mathbf{0} $$ where $$ \mathbf{T}{i}^{(1)}=\left[\mathbf{U}+\mathcal{S}: \mathbf{C}{1}^{-1}:\left(\mathbf{C}{i}-\mathbf{C}_{1}\right)\right]^{-1}
$$